Sains Malaysiana 41(12)(2012): 1643–1649
Mixed Convection Flow about a Solid Sphere Embedded in a
Porous Medium Filled with a Nanofluid
(Aliran Olakan Campuran terhadap Sfera Pejal yang Terbenam
dalam Medium Berliang dengan Nanobendalir)
Leony Tham & Roslinda Nazar*
ABSTRACT
A steady laminar mixed convection boundary layer flow about an isothermal solid sphere embedded in a porous medium
filled with a nanofluid has been studied for both cases of assisting and opposing flows. The transformed boundary layer
equations were solved numerically using an implicit finite-difference scheme. Three different types of nanoparticles,
namely Cu, Al2O3 and TiO2 in water-based fluid were considered. Numerical solutions were obtained for the skin friction
coefficient, the velocity and temperature profiles. The features of the flow and heat transfer characteristics for various
values of the nanoparticle volume fraction and the mixed convection parameters were analyzed and discussed.
Keywords: Boundary layer; mixed convection; nanofluid; porous medium; solid sphere
ABSTRAK
Aliran lapisan sempadan olakan campuran berlamina mantap terhadap sfera pejal isoterma yang terbenam dalam
medium berliang dengan nanobendalir telah dikaji bagi kes aliran membantu dan aliran menentang. Persamaan lapisan
sempadan terjelma diselesaikan secara berangka dengan skema beza terhingga tersirat. Tiga jenis nanozarah dalam
bendalir asas air dipertimbangkan, iaitu Cu, Al2O3 and TiO2. Penyelesaian berangka diperoleh bagi pekali geseran kulit,
profil halaju dan profil suhu. Ciri-ciri aliran dan pemindahan haba bagi pelbagai nilai parameter pecahan isi padu
nanozarah dan parameter olakan campuran dianalisis dan dibincangkan.
Kata kunci: Lapisan sempadan; medium berliang; nanobendalir; olakan campuran; sfera pejal
INTRODUCTION
The convective heat transfer in a fluid saturated porous
media have considerable applications in mechanical,
chemical and civil engineering, including fibrous
insulation, food processing and storage, thermal insulation
of buildings, geophysical systems, the design of pebble
bed nuclear reactors, underground disposal of nuclear
waste, solar power collectors, geothermal applications
and nuclear reactors (Nield & Bejan 2006). Due to the fact
that conventional heat transfer fluids such as oil, water and
ethylene glycol mixture are less productive in transferring
the heat, the thermal conductivity by fluid flow, that is,
the heat transfer coefficient between the heat transfer
medium and the heat transfer surface can be improved
by using nanofluids. Heat transfer by nanofluids is an
innovative technique for improving heat transfer by using
ultra fine solid particles in the fluids. The term nanofluid
was firstly used by Choi (1995) to define the dilution of
nanometer-sized particles (smaller than 100 nm, such as
metals oxides, carbides or carbon nanotubes) in a fluid
(water, ethylene glycol and oil). These nanofluids have
better thermal conductivity and convective heat transfer
coefficient compared with the base fluid only, because of
the diluted nanometer-sized nanoparticles that can easily
flow smoothly through the microchannels (Khanafer et al.
2003) and as such, nanofluids are widely used as coolants,
lubricants and heat exchangers.
The utility of a particular nanofluid for a heat transfer
application can be established by suitably modeling
the convective transport in the nanofluid (Kumar et
al. 2010). Numerical and experimental studies on
nanofluids have been performed, including the study on
thermal conductivity (Kang et al. 2006), separated flow
(Abu-Nada 2008) and convective heat transfer (Jou &
Tzeng 2006). Daungthongsuk and Wongwises (2008)
studied the influence of the thermophysical properties of
nanofluids on the convective heat transfer and summarized
various models used in the literature for predicting the
thermophysical properties of nanofluids. Eastman et al.
(2010) used pure copper nanoparticles of less than 10 nm
in size and achieved a 40% increase in thermal conductivity
for only 0.3% volume fraction of the solid dispersed in
ethylene glycol. Further references on nanofluids can
be found in Das et al. (2007) and in the review paper by
Buongiorno (2006). There are several published papers
on nanofluids that have used the Buongiorno nanofluid
model.
However, in this paper, we will use the nanofluid
model proposed by Tiwari and Das (2007) to study the
present problem of mixed convection boundary layer flow
1644
about a solid sphere embedded in a porous medium filled
with a nanofluid. On the other hand, Nazar et al. (2011)
have considered the mixed convection boundary layer flow
from a horizontal circular cylinder embedded in a porous
medium filled with a nanofluid by extending the papers
by Ahmad and Pop (2010), Cheng (1982), Merkin (1977)
and Nazar et al. (2003). It is worth to point out that this
nanofluid model was very successfully used by Abu-Nada
(2008), Arifin et al. (2011), Abu-Nada and Oztop (2009),
Bachok et al. (2010), Oztop and Abu-Nada (2008) and
Rosca et al. (2012).
BASIC EQUATIONS
Consider the steady mixed convection boundary layer flow
about an impermeable solid sphere of radius a embedded
in a porous medium filled with a nanofluid. It is assumed
that the constant temperature of the surface of the sphere
is Tw, while the constant ambient temperature value is T∞,
where Tw > T∞ for a heated sphere (assisting flow) and
Tw < T∞ for a cooled sphere (opposing flow). It is also
assumed that the velocity of the external flow (inviscisd
flow) or the local free stream velocity is ue(x), where x is
the coordinate measured along the surface of the sphere
starting from the lower stagnation point. The Boussinesq
approximation is employed and the homogeneity and local
thermal equilibrium in the porous medium are assumed.
It is also assumed that the nanoparticles are suspended
in the nanofluid using either surfactant or surface charge
technology. In keeping with the Darcy law and adopting
the nanofluid model proposed by Tiwari and Das (2007),
we obtain the following boundary layer equations for the
problem under consideration in dimensionless form as:
= 0,
(1)
(2)
u
(3)
subject to the boundary conditions
v(x, y) = 0, θ(x, y) = 1
u(x, y) → ue(x) =
θ(x, y) → 0
as
at y = 0, 0 ≤ x ≤ π,
αnf =
(5)
where knf is the effective thermal conductivity of the
nanofluid, kf is the thermal conductivity of the fluid, ks is
the thermal conductivity of the solid, (ρCp)nf is the heat
capacity of the nanofluid, (ρCp)f is the heat capacity of
the fluid and (ρCp)s is the heat capacity of the solid. The
flow is assumed to be slow so that an advective term and
a Forchheimer quadratic drag term do not appear in the
Darcy equation (2). The viscosity of the nanofluid can be
approximated as the viscosity of the base fluid containing
the diluted suspension of fine spherical particles and is
given by Brinkman (1952).
Here λ is the mixed convection parameter, which are
defined as λ = Ra/Pe, with Ra = gKβ(Tw - T∞) a / νf αf,
being the modified Rayleigh number for the porous
medium. It is worth mentioning that λ > 0 for an assisting
flow (Tw > T∞), λ < 0 for an opposing flow (Tw < T∞) and
λ = 0 for the forced convection flow. Integrating Equation
(2) and using the boundary conditions (4), we obtain:
(6)
In order to solve (1), (3) and (6), subject to the
boundary conditions (4), we assumed the following
variables:
nanofluid, ρf and ρs are the density of the fluid and solid
fractions, respectively, μnf is the viscosity of the nanofluid
and g is the magnitude of the gravity acceleration, which
are given by (Oztop & Abu-Nada 2008):
μnf =
(ρCp)nf = (1 – ϕ)(ρCp) + ϕ(ρCp)s,
ψ = xr(x) f (x, y), θ = θ(x, y),
(7)
where ψ is the stream function, which is defined in the
usual way as u = (1/r) ∂ψ/∂y and v = –(1/r)∂ψ/∂x, that
automatically satisfies Equation (1).
Substituting the variables (7) into (3) and (6), and
after some algebra, we obtain the following transformed
equations:
sin x,
y → ∞, 0 ≤ x ≤ π.
(4)
Here u and v are the velocity components along the x
and y axes, respectively, r(x) is the radial distance from the
symmetrical axis to the surface of the sphere, ue(x) is the
local free stream velocity, T is the fluid temperature, ϕ is
the solid volume fraction of the nanofluid or the nanofluid
volume fraction, βf is the thermal expansion coefficient of
the fluid fraction, βs is the thermal expansion coefficient
of the solid fraction, αnf is the thermal diffusivity of the
with the boundary conditions:
(8)
(9)
f (x, y) = 0, θ(x, y) = 1 at y = 0, 0 ≤ x ≤ π,
θ(x, y) → 0 as y → ∞, 0 ≤ x ≤ π
(10)
It can be seen that near the lower stagnation point of
the sphere, i.e. x ≈ 0, (8) and (9) reduce to the following
ordinary differential equations:
(11)
(12)
with the boundary conditions:
f (0) = 0, θ(0) = 1, f´(y) →
θ(y) → 0 as y → ∞
(13)
where primes denote differentiation with respect to y.
Quantity of practical interest is the skin friction coefficient
Cf, which is defined as:
Cf =
(14)
where τw is the surface shear stress, which is given by:
τw =
(15)
Substituting variables (7) into (14) and (15), we
obtain,
(PrPe1/2)Cf =
(16)
RESULTS AND DISCUSSION
Equations (8) and (9), subject to the boundary conditions
(10) have been solved numerically using an efficient
implicit finite-difference scheme known as the Keller-box
method along with the Newton’s linearization technique as
described by Cebeci and Bradshaw (1988). Three different
types of nanoparticles, namely Cu, Al2O3 and TiO2 (with
water as their base fluid), have been considered in this study
and representative results for the skin friction coefficient,
(PrPe1/2)Cf, have been obtained for the following range of
nanoparticle volume fraction ϕ : ϕ = 0, 0.1 and 0.2 (AbuNada & Oztop 2009), at different positions x with various
values of the mixed convection parameter λ. The numerical
solution starts at the lower stagnation point of the sphere,
x ≈ 0 and proceeds round the sphere up to the separation
point. The present results are obtained up to x = 120o only.
We have used data related to thermophysical properties of
the fluid and nanoparticles as given in Oztop and Abu-Nada
(2008) to compute each case of the nanofluid.
1645
Tables 1 and 2 show the values of (PrPe1/2)Cf for ϕ
= 0.1 and 0.2 for nanoparticles Cu at different positions
x and various values of the mixed convection parameter
λ, respectively. It is observed from these tables that the
skin friction coefficients (PrPe1/2)Cf are negative when λ
> 0 and positive when λ < 0 and zero when λ = 0 due to
the definition of (PrPe1/2)Cf given in (16). It is found that
for fixed x and λ, as the value of the nanoparticle volume
fraction ϕ increases from 0 to 0.2, it results in an increase
in the value of the location of the separation point and a
decrease in the magnitude of the skin friction coefficient
(PrPe1/2)Cf and this applies in both the heated sphere (λ >
0) and the cooled sphere (λ < 0) cases.
It is observed from Tables 1 and 2 for the case of the
nanoparticles Cu that the actual value of λ = λs (> 0) which
first gives no separation of boundary layer is difficult to
determine exactly as it has to be found by successive
integration of the equations. However, the numerical
solutions indicate that the value of λs which first gives no
separation lies between −2.09 and −2.10 for ϕ = 0.1 and
between −2.89 and −2.90 for ϕ = 0.2. Meanwhile the value
(negative) of λ = λ0 (< 0) below which a boundary layer
separation is not possible for nanoparticles Cu, are −2.46
for ϕ = 0.1 and −3.3 for ϕ = 0.2. The same trend can be
observed for the cases of nanoparticles Al2O3 and TiO2.
It is found that the boundary layer starts to separate the
fastest (with highest values of λs (> 0)) for the nanoparticles
Cu, followed by Al2O3 and TiO2. This indicates that the
nanoparticles TiO2 delay the start of the boundary layer
separation from the sphere.
Figures 1 and 2 show the skin friction coefficient
(PrPe1/2)Cf for λ = 1 (assisting flow) and λ = −1 (opposing
flow), respectively, with various values of ϕ = 0.0 (regular
Newtonian fluid), 0.1 and 0.2, for the three nanoparticles
considered, namely Cu, Al2O3 and TiO2. It is seen from
these figures that due to the definition of (PrPe1/2)Cf,
the skin friction coefficients are negative when λ > 0
(assisting flow) and positive when λ < 0 (opposing flow).
The negative values of the skin friction coefficient from
Figure 1 corresponds to the surface exerts a drag force on
the fluid and the positive values from Figure 2 implies the
opposite. On the other hand, the parabolic shape of the
curves in Figure 1 for the case of λ = 1 (assisting flow)
implies that the skin friction coefficient is zero at the lower
stagnation point of the sphere, x ≈ 0 and as we proceed
round the sphere, the skin friction coefficient decreases to
its minimum value at around halfway up to the separation
point. Beyond this point, the skin friction coefficient
increases to a finite value at x = 120o. The opposite trends
are observed when λ = −1 (opposing flow) as shown in
Figure 2 with the parabolic curve having a maximum
value. These phenomena are observed for the skin friction
coefficient curves involving a sphere or a circular cylinder.
It is possibly due to the shapes of sphere and circular
cylinder which consequently lead to flow separation. It is
also observed from these figures that the magnitude of the
skin friction coefficient decreases as ϕ increases from 0 to
0.2. Also it is seen that the magnitude of the skin friction
- 2.46
0.0000
0.2096
- 3.30
0.0000
0.2541
x
0o
10o
20o
30o
40o
50o
60o
70o
80o
90o
100o
110o
120o
x
0o
10o
20o
30o
40o
50o
60o
70o
80o
90o
100o
110o
120o
0.0000
0.2683
0.5201
0.7404
0.9145
-3.07
TABLE 2.
0.0000
0.2308
0.4474
0.6369
0.7877
- 2.22
0.0000
0.2341
0.4537
0.6460
0.7995
0.9061
0.9609
0.9629
0.9135
- 2.13
0.0000
0.2347
0.4550
0.6478
0.8017
0.9087
0.9636
0.9657
0.9178
0.8268
0.7026
0.5541
- 2.10
0.0000
0.2349
0.4553
0.6482
0.8023
0.9093
0.9644
0.9664
0.9185
0.8275
0.7037
0.5606
0.4172
- 2.09
0.0000
0.2355
0.4566
0.6500
0.8046
0.9119
0.9671
0.9692
0.9213
0.8302
0.7064
0.5631
0.4158
- 2.0
- 1.0
0.0000
0.1643
0.3184
0.4533
0.5611
0.6360
0.6746
0.6762
0.6429
0.5795
0.4931
0.3928
0.2888
λ
0.0000
0.0913
0.1770
0.2520
0.3119
0.3535
0.3749
0.3758
0.3573
0.3220
0.2739
0.2182
0.1604
- 0.5
-0.0000
-0.1071
-0.2077
-0.2956
-0.3659
-0.4148
-0.4399
-0.4409
-0.4191
-0.3776
-0.3212
-0.2557
-0.1879
0.5
-0.0000
-0.2283
-0.4426
-0.6301
-0.7800
-0.8840
-0.9376
-0.9397
-0.8932
-0.8049
-0.6846
-0.5450
-0.4004
1.0
1. Values of the skin friction coefficient (PrPe1/2) Cf for Cu nanoparticles when ϕ = 0.1 and various values of λ
0.0000
0.2721
0.5275
0.7510
0.9296
1.0536
1.1175
1.1201
1.0656
- 2.94
0.0000
0.2728
0.5289
0.7530
0.9320
1.0563
1.1203
1.1229
1.0674
0.9621
0.8189
0.6515
- 2.90
0.0000
0.2730
0.5292
0.7533
0.9325
1.0569
1.1209
1.1234
1.0680
0.9627
0.8197
0.6551
0.4915
- 2.89
0.0000
0.2448
0.4745
0.6755
0.8362
0.9478
1.0054
1.0079
0.9583
0.8634
0.7341
0.5843
0.4293
- 2.0
0.0000
0.1467
0.2844
0.4049
0.5012
0.5681
0.6026
0.6040
0.5742
0.5175
0.4403
0.3507
0.2579
λ
- 1.0
0.0000
0.0787
0.1525
0.2171
0.2687
0.3046
0.3231
0.3238
0.3079
0.2774
0.2360
0.1880
0.1381
- 0.5
-0.0000
-0.0883
-0.1711
-0.2436
-0.3015
-0.3417
-0.3625
-0.3633
-0.3453
-0.3112
-0.2647
-0.2107
-0.1548
0.5
-0.0000
-0.1853
-0.3593
-0.5115
-0.6331
-0.7176
-0.7611
-0.7628
-0.7251
-0.6533
-0.5557
-0.4424
-0.3250
1.0
2.0
-0.0000
-0.4034
-0.7821
-1.1134
-1.3781
-1.5620
-1.6567
-1.6604
-1.5782
-1.4220
-1.2094
-0.9628
-0.7072
2.0
-0.0000
-1.5932
-3.0885
-4.3966
-5.4420
-6.1681
-6.5417
-6.5559
-6.2311
-5.6138
-4.7738
-3.7994
-2.7901
Values of the skin friction coefficient (PrPe1/2) Cf for Cu nanoparticles when ϕ = 0.2 and various values of λ
TABLE
-0.0000
-1.2207
-2.3663
-3.3686
-4.1696
-4.7259
-5.0122
-5.0232
-4.7745
-4.3015
-3.6580
-2.9116
-2.1383
5.0
-0.0000
-0.5081
-0.9850
-1.4022
-1.7356
-1.9672
-2.0864
-2.0911
-1.9876
-1.7908
-1.5230
-1.2123
-0.8905
5.0
1646
FIGURE
FIGURE
1647
1. Comparison of the skin friction coefficient (PrPe1/2)Cf when λ = 1 (assisting flow) for
various nanoparticles and various values of ϕ (ϕ = 0.0, 0.1 and 0.2)
2. Comparison of the skin friction coefficient (PrPe1/2)Cf when λ = −1 (opposing flow) for
various nanoparticles and various values of ϕ (ϕ = 0.0, 0.1 and 0.2)
coefficient for the regular fluid is higher than the nanofluid.
Among the three nanoparticles, the magnitude of the skin
friction coefficient is the highest for Cu (nanoparticles with
high density and thermal diffusivity), followed by TiO2
and the lowest is Al2O3 (nanoparticles with low thermal
diffusivity). It should be pointed out that nanofluids have
lower skin friction coefficient compared with the base fluid,
which is good to be used as lubricant, due to the suspended
nanoparticles that can stay longer in the base fluid and
the surface area per unit volume of nanoparticles is large.
These two properties can enhance the flow characteristic
of nanofluids.
Figure 3 shows the velocity profiles, f ́(y), at the lower
stagnation point of the sphere, x ≈ 0, of nanoparticles Cu,
when λ = 1, 2 (assisting flow) and λ = −1 (opposing flow),
with the nanoparticle volume fraction ϕ = 0.0, 0.1 and
0.2. It is clear that, as the nanoparticles volume fraction
increases, the nanofluid velocity decreases, however, the
opposite results are observed at λ = −1. Figure 4 shows
the temperature profiles, θ(y), at the lower stagnation
point of the sphere, x ≈ 0, for nanoparticles Cu, when λ
= 1, 2 (assisting flow) and λ = −1 (opposing flow), with
the nanoparticle volume fraction ϕ = 0.0, 0.1 and 0.2.
This figure illustrates this agreement with the physical
1648
3. Velocity profiles f’(y) at the lower stagnation point of the sphere, x ≈ 0, for Cu
nanoparticles with ϕ = 0.0, 0.1 and 0.2 and various values of λ (λ = 2, λ = 1 and λ = −1)
FIGURE
4. Temperature profiles θ(y) at the lower stagnation point of the sphere, x ≈ 0, for Cu
nanoparticles with ϕ = 0.0, 0.1 and 0.2 and various values of λ (λ = 2, λ = 1 and λ = −1)
FIGURE
behavior, that is, when the volume of nanoparticles
increases, the thermal conductivity increases and then
the thermal boundary layer thickness increases. This
is due to the presence of the nanoparticles in the fluids
increases appreciably the effective thermal conductivity
of the fluid and consequently enhances the heat transfer
characteristics (Hady et al. 2012). From all the velocity
and temperature profiles, it is observed that the profiles
satisfy the far field boundary conditions asymptotically,
and as such, this support the validity of the numerical
results obtained.
CONCLUSION
In this paper, we have studied the problem of steady
laminar mixed convection flow over an isothermal sphere
embedded in a porous medium filled with a nanofluid.
We have looked into the effects of the mixed convection
parameter λ, the type of nanoparticles (Al2O3, Cu, TiO2)
and the nanoparticle volume fraction ϕ on the flow and
heat transfer characteristics. The governing non-similar
boundary layer equations were solved numerically using
the Keller-box method. From this study, we could draw the
following conclusions: an increase in the value of ϕ led to a
decrease in the skin friction coefficient, (PrPe1/2)Cf , and to
an increase in the value of λ = λ s (> 0) which first gives no
separation; an increase in the value of ϕ led to a decrease
in the value of λ = λ0 (< 0) below which a boundary layer
solution does not exist and the nanoparticles Cu has the
highest value of the skin friction coefficient, (PrPe1/2)Cf,
compared with the nanoparticles TiO2 and Al2O3. Besides
that, it was found that the type of nanofluid was a key
factor for heat transfer enhancement. The highest values
were obtained when using the Cu nanoparticles.
ACKNOWLEDGEMENT
The authors would like to acknowledge the financial
supports received in the form of the fundamental
research grant scheme (FRGS) and LRGS/TD/2011/
UKM/ICT/03/02 from the Ministry of Higher Education,
Malaysia and UKM-OUP-FST-2012.
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Leony Tham
Fakulti Industri dan Asas Tani
Universiti Malaysia Kelantan
17600 Jeli, Kelantan
Malaysia
Roslindar Nazar*
Pusat Pengajian Sains Matematik
Fakulti Sains dan Teknologi
Universiti Kebangsaan Malaysia
43600 UKM Bangi, Selangor
Malaysia
*Corresponding author; email: [email protected]
Received: 18 May 2012
Accepted: 31 July 2012